Peter J.-S. Shiue, Anthony G. Shannon, Shen C. Huang, Jorge E. Reyes

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 29, 2023, Number 1, Pages 98–129

DOI: 10.7546/nntdm.2023.29.1.98-129

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## Details

### Authors and affiliations

Peter J.-S. Shiue

*Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, NV, 89154, United States of America
*

Anthony G. Shannon

*Warrane College, University of New South Wales
Kensington, NSW 2033, Australia*

Shen C. Huang

*Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, NV, 89154, United States of America
*

Jorge E. Reyes

*Department of Mathematical Sciences, University of Nevada, Las Vegas
Las Vegas, NV, 89154, United States of America
*

### Abstract

A generalized Computation procedure for construction of the Ramanujan-type from a given general cubic equation and a cosine Ramanujan-type identity is developed from detailed analyses of the properties of Ramanujan-type cubic equations. Examples are provided together with cubic Shevelev sums.

### Keywords

- Ramanujan cubic polynomials
- Ramanujan cubic polynomials of the second kind
- Cubic Shevelev sum

### 2020 Mathematics Subject Classification

- 11C08
- 11D25
- 11Y99

### References

- Boyer, C. B., & Merzbach, U. C. (2011).
*A History of Mathematics*. John Wiley & Sons. - Chen, W. Y. C. (2022). Cubic equations through the looking glass of Sylvester.
*The College Mathematics Journal*, 53(5), 396–398. - De Pillis, L. G. (1998). Newton’s cubic roots.
*Gazette of the Australian Mathematical Society*, 25(5), 236–241. - Dresden, G., Panthi, P., Shrestha, A., & Zhang, J. (2019). Cubic polynomials, linear shifts, and Ramanujan simple cubics.
*Mathematics Magazine*, 92(5), 374–381. - Gilbert, L., & Gilbert, J. (2014).
*Elements of Modern Algebra*(8th ed.). Cengage Learning. - Hillman, A. P., & Alexanderson, G. L. (1988).
*A First Undergraduate Course in Abstract Algebra*. Brooks/Cole. - Liao, H.-C., Saul, M., & Shiue, P. J.-S. (in press). Revisiting the general cubic: A

simplification of Cardano’s solution.*The Mathematical Gazette.* - McLeish, J. (1994).
*The Story of Numbers*. Ballantine Books. - Ramanujan, S. (1957).
*Notebooks of Srinivasa Ramanujan*(2 volumes). Tata Institute of Fundamental Research, Bombay. - Shannon, A. G. (1974). The Jacobi–Perron algorithm and Bernoulli’s iteration.
*The Mathematics Student*, 42, 52–56. - Shevelev, V. (2007). On Ramanujan cubic polynomials.
*South East Asian Mathematics and Mathematical Sciences*, 8, 113–122. - Shiue, P. J.-S., Shannon, A. G., Huang, S. C., & Reyes, J. E. (2022). Notes on efficient computation of Ramanujan cubic equations.
*Notes on Number Theory and Discrete Mathematics*, 28(2), 350–375. - Van der Poorten, A. (1996). Notes on Fermat’s last theorem.
*Computers & Mathematics with Applications*, 31(11), 139–139. - Wang, K. (2021). On Ramanujan type identities and Cardano formula.
*Notes on Number Theory and Discrete Mathematics*, 27(3), 155–174. - Wituła, R. (2010). Full description of Ramanujan cubic polynomials.
*Journal of Integer Sequences*, 13, Article 10.5.7. - Wituła, R. (2010). Ramanujan cubic polynomials of the second kind.
*Journal of Integer Sequences*, 13, Article 10.7.5. - Wituła, R. (2012). Ramanujan type trigonometric formulae.
*Demonstratio Mathematica*, 45(4), 779–796.

### Manuscript history

- Received: 25 December 2022
- Revised: 28 February 2023
- Accepted: 2 March 2023
- Online First: 6 March 2023

### Copyright information

Ⓒ 2023 by the Authors.

This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).

## Related papers

- Shiue, P. J.-S., Shannon, A. G., Huang, S. C., & Reyes, J. E. (2022). Notes on efficient computation of Ramanujan cubic equations.
*Notes on Number Theory and Discrete Mathematics*, 28(2), 350–375. - Wang, K. (2021). On Ramanujan type identities and Cardano formula.
*Notes on Number Theory and Discrete Mathematics*, 27(3), 155–174.

## Cite this paper

Shiue, P. J.-S., Shannon, A. G., Huang, S. C., & Reyes, J. E. (2023). A generalized computation procedure for Ramanujan-type identities and cubic Shevelev sum. *Notes on Number Theory and Discrete Mathematics*, 29(1), 98-129, DOI: 10.7546/nntdm.2023.29.1.98-129.